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Formula form
Ax² + Bx + C = 0
Allow negative discriminant
Discriminant (Δ)
Your function
Standard form: f(x) = Ax² + Bx + C
Vertex form: f(x) = A(x - H)² + K
Factored form: f(x) = A(x - x₁)(x - x₂)

Parabola with a > 0 and real roots

If you need to solve an equation of form Ax² + Bx + C = 0, this quadratic formula calculator is here to help you. With just a few clicks you will be able to solve even the most difficult problem. This article describes in detail what is the quadratic formula and what do the symbols A, B and C stand for. It also explains how to solve quadratic equations which have a negative determinant and no real roots.

What is the quadratic formula?

The quadratic formula is the solution of a second degree polynomial equation of the following form:

Ax² + Bx + C = 0

If you are able to bring your equation to this form, it means that it can be solved with the quadratic formula. The solution of this equation is also called a root of an equation.

The quadratic formula is as follows:

x = (-B ± √Δ)/2A


Δ = B² - 4AC

Using this formula, you can find the solutions of any quadratic equation. Note that there are three possible options of obtaining a result:

  • The quadratic equation has two unique roots when Δ > 0. Then, the first solution of the quadratic formula is x₁ = (-B + √Δ)/2A and the second is x₂ = (-B - √Δ)/2A.
  • The quadratic equation has only one root when Δ = 0. The solution is equal to x = -B/2A. It is sometimes called a repeated or double root.
  • The quadratic equation has no real solutions for Δ < 0.

You can also graph the function y = Ax² + Bx + C. It's shape is a parabola, and the roots of the quadratic equation are the x-intercepts of this function.

Coefficients of a quadratic equation

A, B and C are the coefficients of the quadratic equation. They are all real numbers not dependent on x. If A = 0, then the equation is not quadratic, but linear.

If B² < 4AC, then the determinant Δ will be negative. It means that such an equation has no real roots.

How to use the quadratic formula solver

  1. Write down your equation. Let's assume it is 4x² + 3x - 7 = -4 - x.
  2. Bring the equation to the form Ax² + Bx + C = 0. In this example, we will do it in the following steps:

4x² + 3x - 7 = -4 - x

4x² + (3+1)x + (-7+4) = 0

4x² + 4x - 3 = 0

  1. Calculate the determinant.

Δ = B² - 4AC = 4² - 4*4*(-3) = 16 + 48 = 64.

  1. Decide whether the determinant is greater, equal or lower than 0. In our case, the determinant is greater than 0, what means that there are two unique roots of this equation.
  2. Calculate the two roots using the quadratic formula.

x₁ = (-B + √Δ)/2A = (-4 +√64) / (2*4) = (-4+8) / 8 = 4/8 = 0.5

x₂ = (-B - √Δ)/2A= (-4 -√64) / (2*4) = (-4-8) / 8 = -12/8 = -1.5

  1. The roots of your equation are x₁ = 0.5 and x₂ = -1.5.

You can also simply type the values of A, B and C into our quadratic equation calculator and let it perform all calculations for you.

Make sure you have written down the correct number of digits using our significant figures calculator.

Solving quadratic equations with a negative determinant

Even though the quadratic formula calculator indicates when the equation has no real roots, it is possible to find the solution of a quadratic equation with a negative determinant. These roots will be complex numbers.

Complex numbers have a real and imaginary part. The imaginary part is always equal to the number i = √(-1) multiplied by a real number.

The quadratic formula remains the same in this case.

x = (-B ± √Δ)/2A

Notice that, as Δ < 0, the square root of the determinant will be an imaginary value. Hence

Re(x) = -B/2A

Im(x) = ± (√Δ)/2A

Extra resources

If after learning all this about solving quadratic equations, you still want more math. We recommend you to check out the Computer Technology For Math Excellence's webpage. They have a huge collection of resources to learn all about math, with especial attention to the Common Core curriculum.

Bogna Haponiuk and Dominik Czernia, PhD candidate

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